Bayesian Model Selection in High-Dimensional Settings.

Area of Science:

• Statistics
• Computational Statistics
• Bayesian Inference

Background:

• Standard Bayesian model selection methods often underperform compared to penalized likelihood approaches.
• Existing methods lack consistency guarantees, especially when the number of covariates is large relative to observations.

Purpose of the Study:

• To develop a modified Bayesian model selection procedure that is competitive with penalized likelihood methods.
• To establish theoretical consistency properties for the proposed method in linear models.
• To provide accurate posterior probability estimates for selected models.

Main Methods:

• Modification of standard Bayesian model selection by incorporating nonlocal prior densities on model parameters.
• Theoretical analysis of consistency properties under the condition that the number of covariates (p) is bounded by the number of observations (n).
• Simulation studies to compare performance against established penalized likelihood methods.

Main Results:

• The proposed nonlocal prior Bayesian model selection procedures demonstrate consistency in linear models.
• These procedures achieve accurate estimation of the posterior model probabilities.
• Simulation results indicate performance comparable to or exceeding penalized likelihood methods across various settings.

Conclusions:

• The novel Bayesian approach with nonlocal priors offers a consistent and accurate alternative for model selection in linear models.
• This method extends consistency properties to settings where p <= n, a significant theoretical advancement.
• The proposed procedures provide a robust framework for both model identification and probability assessment.